big league vs league

big league and league both are nouns.

Word	Noun	Adjective	Verb	Adverb
big league	Yes	No	No	No
league	Yes	No	Yes	No

As nouns, league is a hypernym of big league; that is, league is a word with a broader meaning than big league:

big league: the most important league in any sport (especially baseball)
league: an association of sports teams that organizes matches for its members

Other hypernyms of big league include conference.

big league (noun)	league (noun)
the most important league in any sport (especially baseball)	an association of states or organizations or individuals for common action
	an association of sports teams that organizes matches for its members
	an obsolete unit of distance of variable length (usually 3 miles)

big league (verb)	league (verb)
	unite to form a league